Adaptive Out-Orientations with Applications

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum outdegree is proportional to the arboricity α of the graph, in, either, an amortised update time of O(log² nlog α), or a worst-case update time of O(log³ nlog α). On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either O(log nlog α), amortised, or O(log² nlog α), worst-case, for the problem of maintaining an edge-orientation with at most O(α + log n) out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in n and α. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a (1 + ε) approximation of the maximum subgraph density, ρ, of the dynamic graph. Our algorithms have update times of O(ε⁻⁶ log³ nlog ρ) worst-case, and O(ε⁻⁴ log² nlog ρ) amortised, respectively. We may output a subgraph H of the input graph where its density is a (1 + ε) approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the O(ε⁻⁶ log⁴ n) algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an O(ε⁻⁶ log³ nlog α) worst-case update time algorithm for maintaining a (1 + ε)OPT + 2 approximation of the optimal out-orientation of a graph with adaptive arboricity α, improving the O(ε⁻⁶α² log³ n) algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into O(α) forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, ∆ + 1 colouring, and matrix vector multiplication. All update times are worst-case O(α + log² nlog α), where α is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time O(α² + log² n), and by Neiman and Solomon from STOC 2013 runs in time O(√m). We give improved running times whenever the arboricity α ∈ ω(log n√log log n).